Abstract
We prove Kontsevich’s homological mirror symmetry conjecture for certain mirror pairs arising from Batyrev–Borisov’s ‘dual reflexive Gorenstein cones’ construction. In particular we prove HMS for all Greene–Plesser mirror pairs (i.e., Calabi–Yau hypersurfaces in quotients of weighted projective spaces). We also prove it for certain mirror Calabi–Yau complete intersections arising from Borisov’s construction via dual nef partitions, and also for certain Calabi–Yau complete intersections which do not have a Calabi–Yau mirror, but instead are mirror to a Calabi–Yau subcategory of the derived category of a higher-dimensional Fano variety. The latter case encompasses Kuznetsov’s ‘K3 category of a cubic fourfold’, which is mirror to an honest K3 surface; and also the analogous category for a quotient of a cubic sevenfold by an order-3 symmetry, which is mirror to a rigid Calabi–Yau threefold.
Highlights
1.1 Toric mirror constructionsOne of the first constructions of mirror pairs of Calabi–Yau varieties was due to Greene and Plesser [28]
They considered Calabi–Yau hypersurfaces in quotients of weighted projective spaces. They were interested in the threedimensional case, but their construction works just as well in any dimension. Batyrev generalized this to a construction of mirror pairs of Calabi–Yau hypersurfaces in toric varieties [8]
Batyrev and Borisov succeeded in generalizing their constructions to include these generalized Calabi–Yau varieties. They constructed mirror pairs of Landau–Ginzburg models, depending on dual pairs of ‘reflexive Gorenstein cones’ [9]. They showed that a reflexive Gorenstein cone equipped with a ‘complete splitting’ determines a Calabi–Yau complete intersection in a toric variety, which should be equivalent to the Landau–Ginzburg model via the ‘Landau–Ginzburg/Calabi–Yau correspondence’
Summary
One of the first constructions of mirror pairs of Calabi–Yau varieties was due to Greene and Plesser [28]. They considered Calabi–Yau hypersurfaces in quotients of weighted projective spaces They were interested in the threedimensional case, but their construction works just as well in any dimension. Batyrev generalized this to a construction of mirror pairs of Calabi–Yau hypersurfaces in toric varieties [8]. Borisov generalized Batyrev’s construction to encompass mirror pairs of Calabi–Yau complete intersections in toric varieties [13]. They constructed mirror pairs of Landau–Ginzburg models, depending on dual pairs of ‘reflexive Gorenstein cones’ [9] They showed that a reflexive Gorenstein cone equipped with a ‘complete splitting’ determines a Calabi–Yau complete intersection in a toric variety, which should be equivalent to the Landau–Ginzburg model via the ‘Landau–Ginzburg/Calabi–Yau correspondence’. We remark that generalized Greene–Plesser mirrors include all Greene–Plesser mirrors, and work through the case of the quartic surface and its mirror (in the ‘reverse’ direction from that considered in [55]); we consider some examples which do not arise from the Greene–Plesser construction, including the rigid Calabi–Yau threefold mentioned above, as well as a certain K 3 surface which is mirror to Kuznetsov’s ‘K 3 category associated to the cubic fourfold’ [38]
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
